| L(s) = 1 | + 0.346i·3-s + 2.94i·5-s + 4.18i·7-s + 2.87·9-s + 2.89i·11-s − 3.88·13-s − 1.02·15-s + (−3.65 + 1.90i)17-s − 2.51·19-s − 1.44·21-s − 6.02i·23-s − 3.69·25-s + 2.03i·27-s − 2.25i·29-s + 4.79i·31-s + ⋯ |
| L(s) = 1 | + 0.200i·3-s + 1.31i·5-s + 1.58i·7-s + 0.959·9-s + 0.871i·11-s − 1.07·13-s − 0.263·15-s + (−0.887 + 0.461i)17-s − 0.575·19-s − 0.316·21-s − 1.25i·23-s − 0.739·25-s + 0.392i·27-s − 0.418i·29-s + 0.860i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.231796128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.231796128\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (3.65 - 1.90i)T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 0.346iT - 3T^{2} \) |
| 5 | \( 1 - 2.94iT - 5T^{2} \) |
| 7 | \( 1 - 4.18iT - 7T^{2} \) |
| 11 | \( 1 - 2.89iT - 11T^{2} \) |
| 13 | \( 1 + 3.88T + 13T^{2} \) |
| 19 | \( 1 + 2.51T + 19T^{2} \) |
| 23 | \( 1 + 6.02iT - 23T^{2} \) |
| 29 | \( 1 + 2.25iT - 29T^{2} \) |
| 31 | \( 1 - 4.79iT - 31T^{2} \) |
| 37 | \( 1 - 7.22iT - 37T^{2} \) |
| 41 | \( 1 - 1.10iT - 41T^{2} \) |
| 43 | \( 1 + 0.255T + 43T^{2} \) |
| 47 | \( 1 + 1.67T + 47T^{2} \) |
| 53 | \( 1 - 3.70T + 53T^{2} \) |
| 61 | \( 1 - 4.76iT - 61T^{2} \) |
| 67 | \( 1 - 4.32T + 67T^{2} \) |
| 71 | \( 1 + 13.9iT - 71T^{2} \) |
| 73 | \( 1 + 8.35iT - 73T^{2} \) |
| 79 | \( 1 + 7.38iT - 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 4.00T + 89T^{2} \) |
| 97 | \( 1 - 6.67iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.956284875682656729769687869112, −8.122123034905796882050306865049, −7.26140006819548031698691330600, −6.60386335825550949695406553189, −6.19911921666617540913076517904, −4.92737393767255286665376764499, −4.52823588301852705450375015516, −3.30707429115795304997337746964, −2.38749595971352133229675703510, −2.01632595630363431662191780704,
0.36908242906343977408657493716, 1.12299964412198227106555680499, 2.16753619629984580046512934392, 3.64602892414409732964618445109, 4.26900875491654377818010996957, 4.85267868206745676879732328036, 5.68077840927195401274550978211, 6.82113362938513389805165773760, 7.28856769508938731727980184045, 7.939017202307429723124519921505
